Table Of ContentPREFACE
The present volume contains the proceedings of the International
Conference on "Differential Geometric Methods in Mathematical Physics'
held at the Technical University of Clausthal in July 1978. The con-
ference continues the tradition of the Bonn conference series devoted
to an exchange between physics and mathematics, particularly in the
fields of geometry and topology applied to gravitation, particle
physics and quantization methods. According to their tradition these
conferences are not only an occasion to communicate physical and
mathematical results and their interrelation, but also to report
on mathematical structures and techniques which could help to under-
stand and to unite experimental results and, using the momentum of
successful application of a mathematical structure in physics, to
develop and to extend it.
The conference, organized in cooperation with K. Bleuler, Bonn, and
W.H. Greub, Toronto, was centered around the following topics:
Quantization Methods and Special Quantum Systems
-
geometric quantization, vectorfield quantization, quantization
of stochastic phase spaces, dynamics of magnetic monopoles,
spectrum generating groups
Gauge Theories
-
phase space of the classical Yang-Mills equation, nonlinear
- models, gauging geometrodynamics, exceptional gauge groups
- Elliptic Operators, Spectral Theory and Applications
the Atiyah-Singer theorem applied to quantum-field theory,
spectral theory applied to phase transitions
Geometric Methods and Global Analysis
-
systems on non-Hausdorff spaces and on non-Euclidean spaces,
Weyl geometry, Lorentz manifolds, manifolds of embeddings.
The contributions in this volume cover almost all the material pre-
sented in the conference; one paper is included through its abstract.
The responsibili£y for the final preparation of the manuscripts for
the printing was in the hands of the editor. I thank B. Angermann
for his assistance and W. Weihrauch for typing the manuscripts.
VI
The organizers wish to express their gratitude to the Volkswagen-
stiftung and to the Technische Universit~t Clausthal for their most
generous financial help. They are indebted to Mrs. Jutta M0ller for
the excellent and invaluable work as conference secretary, to all
lecturers and participants, and to the members of the Clausthal
Institute for Theoretical Physics whose effort made the conference
what it was: lively and stimulating, i.e., successful.
H. Doebner
TABLE OF CONTENTS
I. QUANTIZATION METHODS AND SPECIAL QUANTUM SYSTEMS
HESS, H. On a Geometric Quantization Scheme
Generalizing Those of Kostant-Souriau
and Czyz ................................. I
SNIATYCKI, J. Further Applications of Geometric
Quantization ............................. 36
PASEMANN, F.B. General Vector Field Representations
of Local Heisenberg Systems .............. 38
ALI, S.T. Aspects of Relativistic Quantum
Mechanics on Phase Space ................. 49
PETRY, H.R. On the Confinement of Magnetic
Poles .................................... 77
BOHM, A. &
TEESE, R.B. SU(3) and SU(4) as Spectrum-
Generating Groups ........................ 87
2. GAUGE THEORIES
SEGAL, I.E. The Phase Space for Yang-Mills
Equations ................................ 101
FORGER, M. Instantons in Nonlinear o-Models,
Gauge Theories and General Relativity .... 110
MIELKE, E.W. Gauge-Theoretical Foundation of Color
Geometrodynamics ......................... 135
BIEDENHARN, L.C. &
HORWITZ, L.P. Non-Associative Algebras and Exceptional
Gauge Groups ....... ...................... 152
IV
3. ELLIPTIC OPERATORS, SPECTRAL THEORY AND PHYSICAL APPLICATIONS
ROMER, H. Atiyah-Singer Index Theorem and Quantum
Field Theory ............................. 167
RASETTI, M. Topological Concepts in Phase Transition
Theory ................................... 212
4. GEOMETRIC METHODS AND GLOBAL ANALYSIS
DOMIATY, R.Z. Life Without T 2 .......................... 251
SLAWIANOWSKI, J.J. Affine Model of Internal Degrees of
Freedom in a Non-Euclidean Space ......... 259
HENNIG, J.D. Jet Bundles and Weyl Geometry ............ 280
GREUB, W.H. Line Fields and Lorentz Manifolds ........ 290
BINZ, E. &
FISCHER, H.R. The Manifold of Embeddings of a
Closed Manifold .......................... 310
List of participants
S.T.AIi, Toronto, Canada G.Karrer, ZHrich, Switzerland
E.Aguirre, Madrid, Spain ,ymrcK.R.S Riyad, Saudi Arabia
B.Angermann, Clausthal, FRG D.Krausser, TU Berlin, FRG
A.O.Barut, Boulder, USA K.Just, Tuscon, USA
L.C.Biedenharn, Durham, USA W.LHcke, Clausthal, FRG
E.Binz, Mannheim, FRG E.W.Mielke, Kiel, FRG
.K Bleuler, Bonn, FRG F.B.Pasemann, Clausthal, GAIF
P.Cam~bell, Lancaster, England H.R.Petry, Bonn, FRG
P.Cotta-Ramusino, Mailand, Italy .T Rasetti, Turin, Italy
H.D.Doebner, Clausthal, FRG H.Rdaer, Cern, Switzerland
R.Dcmiaty, Graz, Austria I.E.Segal, Cambridge, USA
,lhtirD.K Starnberg, FRG H.J.Schmidt, OsnabrHck, FRG
M.Forger, FU Berlin, FRG A.Schober, TU Berlin, FRG
P.L.Garcia-Perez, Salamanca, Spain J.Slawianowski, Warschau, Poland
G.Gerlich, Braunschweig, FRG J.Sniatycki, Calgary, Canada
W.Greub, Toronto, Canada .J Tarski, Clausthal, FRG
G.C.Hegerfeldt, ,negnittS~C FRG R.B.Teese, Austin, USA
K.-E.Hellwig, TU Berlin, FRG R.Wilson, ,nehcniiM FRG
J.D.Hennig, K~in, FRG J.-E.Werth, Clausthal, FRG
H.HeB, FU Berlin, FRG
Y.Ingvason, ,negnittS/C FRG
and other participants frc~ the
Technical University of Clausthal.
On a geometric quantization scheme generalizing
those of Kostant-Souriau and Czyz
Harald Hess
Freie Universit~t Berlin
FB 20, WE 4
Arnimallee 3
D-IOOO Berlin 33
Abstract: A quantization method (strictly generalizing the Kostant °
Souriau theory) is defined, which may be applied in some cases where
both Kostant-Souriau prequantum bundles and metaplectic structures
do not exist. It coincides with the Czyz theory for compact K~hler
manifolds with locally constant scalar curvature. Quantization of
dynamical variables is defined without use of intertwining operators,
extending either the Kostant map or some ordering rule like that of
Weyl or Born-Jordan.
O. Introduction:
The aim of this article is to present a new method for geometric
quantization extending that of Kostant-Souriau in two respects.
First, the Kostant-Souriau theory cannot be applied to classical
phase spaces with non-vanishing second Stiefel-Whitney class, since
in this case metaplectic structures and half-forms do not exist.
This problem arises for quantization of energy surfaces of the
n-dimensional harmonic oscillator, where the reduced phase space is
pn-1(~), for odd n (n > I). In case of the Schr~dinger energy levels,
even Kostant-Souriau prequantum bundles do not exist, which has been
recognized by Czyz 7, 8who invented another geometric quantization
theory for compact K~hler manifolds without the mentioned disadvantages.
As in the Kostant-Souriau theory, the quantizing Hilbert space there
is also built from sections in a complex line bundle, but the latter
is directly chosen to satisfy some basic postulates, while in the
Kostant-Souriau theory it is the tensor product of the prequantum
bundle and the bundle of the half-forms. In addition, the connection
on the line bundle is an ordinary one in the Czyz theory, while in the
Kostant-Souriau theory it is only a partial connection, which can be
evaluated only along the respective polarization.
Secondly, the Kostant-Souriau theory does not yield self-adjoint
operators for moderately general functions on phase space. In fact,
the quantizing operators are not even formally self-adjoint when the
function in question is (roughly) a polynomial in the momentum
variables of order strictly greater than ,2 see Kostant 18 for this
statement.
To cure these defects,the basic philosophy of our new approach
is to examine closely the relationship between conventional quantum
theory and geometric quantization. The latter will be obtained from
the former applied in the tangent spaces, being locally curved and
globally twisted. In sophisticated terms, conventional quantum
theory deals with symplectic vector spaces and irreducible weyl systems
(representations of the CCR in exponential form) thereon. The global
twisting has to be performed with automorphisms of the given Weyl
system. We denote the automorphism group by MpC(2n,~). It has been
studied extensively by A. Weyl 32 , and it is the precise symplectic
analogue of the (extended) orthogonal spinor group known as SpinC(2n),
cf. 2 .
How the global twisting has to be done will be coded in a principal
MpC(2n, ~)-bundle P adapted to the given 2n-dimensional symplectic
manifold (M,~). To construct differential operators on complex line
bundles arising from P, the latter should be equipped with an
ordinary connection. Only one part of this connection will be fairly
uniquely determined by (M,~), the other one will be yielded by
polarizations. Existence and classification of such MpC(2n,~ )-bundles
with the polarization-independent part of the connection is discussed
in section ,I where it is also shown how to get these data when
Kostant-Souriau prequantum bundles and metaplectic structures are
given.
In section ,2 it will be seen that two transverse polarizations
determine a unique torsion-free symplectic connection. Together with
the data of section ,I it allows to construct complex line bundles
with connection, generalizing those of the Kostant-Souriau and Czyz
theories. The construction is done in section 3 by a procedure of
reducing the structure group of the principal MpC(2n, ~)-bundle and
subsequent building associated bundles. It is somewhat complicated,
but very similar to the way of getting half-form bundles from
metaplectic structures in the Kostant-Souriau theory. The complex
line bundles yielded by this procedure satisfy the dogma of having
1~2 + ½ cI(TM,~) as their first (real) Chern class, when e
c I (TM,~) is a symplectic invariant of ,M( ~ .)
The r~le of this dogma is to some extent clarified in section 4,
where we indicate how to assign differential operators on the above
line bundles to certain functions on M. Such a map will be obtained
by generalizing ordering rules of conventional quantum theory, like
those of Weyl or Born-Jordan, via replacing ordinary differentiations
by the connections from above. One of the possible ordering rules
gives a map similar to that considered by Kostant 18 . Note that we
don't need any kind of intertwining operators as long as the functions
on the phase space are of a special type.
Our approach is a strict extension only of the Kostant-Souriau theory.
In contrast, it generalizes the Czyz theory just for some restricted
class of K~hler manifolds, which contains all examples explicitly
investigated by Czyz.
The results of section 4 and most of those in section 3 have not
been part of the original conference talk. The material presented
here will be treated in more detail in the author's doctor thesis 133 ;
see also 9 for another view of a special case.
.I Prequantization
The notion of prequantization used here is not quite identical to
that already established in the literature, but is meant to refer to
all polarization-independent constructions. In the case of the
Kostant-Souriau theory it includes both Kostant-Souriau prequantum
bundles and metaplectic frame bundles, which we shall briefly recall
in the beginning. All concepts of the Kostant-Souriau theory will
henceforth be specified by the prefix KS.
Then we indicate why the fundamental structure group must be
MpC(2n, )RI rather than U(1) x Mp(2n, )RI as in the KS-theory. Next, we
give the definition of a (generalized) prequantum bundle with structure
group MpC(2n, IR), and derive two existence criteria for them, one in
terms of the cohomology classes - ~ and cI(TM,~). Also, an
equivalence relation between such (generalized) prequantum bundles
will be defined, and the corresponding equivalence classes turn out to
be in bijection with the elements of HI ,M( U(1)) or the empty set.
Finally, we show how to construct (generalized) prequantum bundles
from given KS-prequantum bundles and metaplectic structures. The
latter are a superfluous degree of freedom in the sense that, up to
equivalence, every (generalized) prequantum bundle can be obtained
in this way from an arbitrary metaplectic structure.
(M,~) will always denote a fixed 2n-dimensional symplectic
manifold. All bundles have base M, and all bundle morphisms are
supposed to induce the identity on M, if not stated otherwise.
Given any Lie group G, we denote by G the corresponding sheaf of
(germs of) C ~ functions on M with values in G. We use Cech cohomology
with coefficients in sheaves of not necessarily abelian groups,
referring to 10 , 30 .
1.1. Definition:
A KS-prequantum bundle (over~) is a principal U(1)-bundle
VL: ~M equipped with a principal connection ~ satisfying
)1( V'I-U,C ~ = ~TI" L~
Given another KS-prequantum bundle (~', ~'), both will be called
equivalent if there exists a principal bundle morphism ~ : ~ ~ ~'
such that
(2)
=
KS-prequantum bundles can also be viewed as hermitian complex line
bundles L equipped with a (linear) connection LV such that
curvLv = i
Obviously, the first (real) Chern class of every KS-prequantum
bundle (over ~ ) is - ~ , in particular it is (the image of) an
integral class. Moreover, we have the well-known existence criterion
and classification 20 , 28 , 33 :
1.2. Theorem:
There exists a KS-prequantum bundle over ~ if and only if the
following equivalent conditions are satisfied
)3( - ~ & ,M( )RI is an integral class
)4( the class H2(M, exp)i~ 6 H2(M, U(1)) vanishes.
Equivalence of )3( and )4( is easily seen from the cohomology
sequence induced by the exact sequence of groups
O ~ • 2 ~i • R/J exp ; U(1) ) 0 "
1.3. Theorem:
The group HI(M, U(1)) operates in a simply transitive manner on the
set of equivalence classes of KS-prequantum bundles over ~ . In
particular this set is either void or in bijection with HI(M, U(1)).
Both of these theorems have proven to be physically significant.
Indeed, the existence condition e.g. restricts the values of quantized
I
spin to integer multiples of ~ 23 , 28 , while the classification
provides for different (Bose and Fermi) quantizations for systems
composed of a number of indistinguishable subsystems 28 or for the
3-dimensional rotator 23 .
Now consider a central extension of Lie groups
)5( O ~ C > G > G > 0
and a principal G-bundle P.
1.4. Definition:
A ~ -lifting of P is a principal ~-bundle ~ together with a
-equivariant principal bundle morphism ~: ~ ~ P.
Given another ~-lifting (~', ~') of P, both will be called
equivalent if there exists a principal bundle morphism ~ : ~ -- ~'
such £hat the diagram
commutes.
Let us identify the isomorphism class P with the corresponding
cohomology class in H I ,M( )_G induced by a system of transition
functions of P. Further, consider the cohomology sequence induced by
the sequence of sheaves of C~-functions corresponding to )5(
)7( ) H I (M,C) > H I (M,G) ~ H I )G_,M( H 2(M,C) .
Calling w~(P) := ~1p the ~ -obstruction class of P, we have the
well-known existence criterion due to 10 .
1.5. Theorem:
P admits a ~-lifting if and only if the cohomology class w~(P) e
H 2(M,C) vanishes.
